So if I add a book from a new author why this original spelling gets sticked
It should be updated as I enter a new sort field value, shouldn't it?

Calibre has a default IGNORE_CASE turned on for various areas and tries to handle both case sensitive and case insensitive operating systems. You're running into one of the problems caused by the ignore case default. Either do what itimpi suggested, or just change something other than the case of the new author_sort, followed by changing it again. To avoid these issues, never make any change that involves only changing the case.